The Stacks project

94.11 Quasi-coherent modules

At this point we can apply the general definition of a quasi-coherent module to the situation discussed in this chapter.

Definition 94.11.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. A quasi-coherent module on $\mathcal{X}$, or a quasi-coherent $\mathcal{O}_\mathcal {X}$-module is a quasi-coherent module on the ringed site $(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$ as in Modules on Sites, Definition 18.23.1. The category of quasi-coherent sheaves on $\mathcal{X}$ is denoted $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

If $\mathcal{X}$ is an algebraic stack, then this definition agrees with all definitions in the literature in the sense that $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is equivalent (modulo set theoretic issues) to any variant of this category defined in the literature. For example, we will match our definition with the definition in [Definition 6.1, olsson_sheaves] in Cohomology on Stacks, Lemma 94.11.6. We will also see alternative constructions of this category later on.

In general (as is the case for morphisms of schemes) the pushforward of quasi-coherent sheaf along a $1$-morphism is not quasi-coherent. Pullback does preserve quasi-coherence.

Lemma 94.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal {Y}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ preserves quasi-coherent sheaves.

Proof. This is a general fact, see Modules on Sites, Lemma 18.23.4. $\square$

It turns out that quasi-coherent sheaves have a very simple characterization in terms of their pullbacks. See also Lemma 94.11.6 for a characterization in terms of restrictions.

Lemma 94.11.3. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent sheaf on $(\mathit{Sch}/U)_{fppf}$ for every object $x$ of $\mathcal{X}$ with $U = p(x)$.

Proof. By Lemma 94.11.2 the condition is necessary. Conversely, since $x^*\mathcal{F}$ is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it is sufficient directly from the definition of a quasi-coherent sheaf (and the fact that the notion of being quasi-coherent is an intrinsic property of sheaves of modules, see Modules on Sites, Section 18.18). $\square$

Lemma 94.11.4. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of modules on $\mathcal{X}$. The following are equivalent

  1. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1,

  2. $\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1, and

  3. $\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$ in the sense of Definition 94.11.1.

Proof. Assume either (1), (2), or (3) holds. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Recall that $x^*\mathcal{F} = x^{-1}\mathcal{F}$ is just the restriction to $\mathcal{X}/x = (\mathit{Sch}/U)_\tau $ where $\tau = fppf$, $\tau = {\acute{e}tale}$, or $\tau = Zar$, see Section 94.9. By the definition of quasi-coherent modules on a ringed site this restriction is quasi-coherent provided $\mathcal{F}$ is. By Descent, Proposition 35.8.11 we see that $x^*\mathcal{F}$ is the sheaf associated to a quasi-coherent $\mathcal{O}_ U$-module and is therefore a quasi-coherent module in the fppf, étale, and Zariski topology; here we also use Descent, Lemma 35.8.1 and Definition 35.8.2. Since this holds for every object $x$ of $\mathcal{X}$, we see that $\mathcal{F}$ is a sheaf in any of the three topologies. Moreover, we find that $\mathcal{F}$ is quasi-coherent in any of the three topologies directly from the definition of being quasi-coherent and the fact that $x$ is an arbitrary object of $\mathcal{X}$. $\square$

Although there is a variant for the Zariski topology, it seems that the étale topology is the natural topology to use in the following definition.

Definition 94.11.5. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal {X}$-modules. We say $\mathcal{F}$ is locally quasi-coherent1 if $\mathcal{F}$ is a sheaf for the étale topology and for every object $x$ of $\mathcal{X}$ the restriction $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ is a quasi-coherent sheaf. Here $U = p(x)$.

We use $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ to indicate the category of locally quasi-coherent modules. We now have the following diagram of categories of modules

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar[r] \ar[d] & \textit{Mod}(\mathcal{O}_\mathcal {X}) \ar[d] \\ \textit{LQCoh}(\mathcal{O}_\mathcal {X}) \ar[r] & \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) } \]

where the arrows are strictly full embeddings. It turns out that many results for quasi-coherent sheaves have a counter part for locally quasi-coherent modules. Moreover, from many points of view (as we shall see later) this is a natural category to consider. For example the quasi-coherent sheaves are exactly those locally quasi-coherent modules that are “cartesian”, i.e., satisfy the second condition of the lemma below.

Lemma 94.11.6. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal {X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if the following two conditions hold

  1. $\mathcal{F}$ is locally quasi-coherent, and

  2. for any morphism $\varphi : x \to y$ of $\mathcal{X}$ lying over $f : U \to V$ the comparison map $c_\varphi : f_{small}^*\mathcal{F}|_{V_{\acute{e}tale}} \to \mathcal{F}|_{U_{\acute{e}tale}}$ of (94.9.4.1) is an isomorphism.

Proof. Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf for the fppf topology, hence a sheaf for the étale topology. Moreover, any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ are quasi-coherent. This proves $\mathcal{F}$ is locally quasi-coherent. Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. We have seen that $\mathcal{X}/y = (\mathit{Sch}/V)_{fppf}$. By Descent, Proposition 35.8.11 it follows that $y^*\mathcal{F}$ is the quasi-coherent module associated to a (usual) quasi-coherent module $\mathcal{F}_ V$ on the scheme $V$. Hence certainly the comparison maps (94.9.4.1) are isomorphisms.

Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2). Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote $\mathcal{F}_ V$ the quasi-coherent module on the scheme $V$ corresponding to the restriction $y^*\mathcal{F}|_{V_{\acute{e}tale}}$ which is quasi-coherent by assumption (1), see Descent, Proposition 35.8.11. Condition (2) now signifies that the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ for $x$ over $y$ are each isomorphic to the (étale sheaf associated to the) pullback of $\mathcal{F}_ V$ via the corresponding morphism of schemes $U \to V$. Hence $y^*\mathcal{F}$ is the sheaf on $(\mathit{Sch}/V)_{fppf}$ associated to $\mathcal{F}_ V$. Hence it is quasi-coherent (by Descent, Proposition 35.8.11 again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by Lemma 94.11.3. $\square$

Lemma 94.11.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{Y}_{\acute{e}tale}, \mathcal{O}_\mathcal {Y}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ preserves locally quasi-coherent sheaves.

Proof. Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$. Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$. The restriction $x^*f^*\mathcal{G}|_{U_{\acute{e}tale}}$ equals $(f \circ x)^*\mathcal{G}|_{U_{\acute{e}tale}}$ hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$. $\square$

Lemma 94.11.8. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.

  1. The category $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$.

  2. The category $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ is abelian with kernels and cokernels computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, in other words the inclusion functor is exact.

  3. Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ if two out of three are locally quasi-coherent so is the third.

  4. Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$.

  5. Given $\mathcal{F}, \mathcal{G}$ in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{\acute{e}tale}$ the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is an object of $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$.

Proof. Each of these statements follows from the corresponding statement of Descent, Lemma 35.8.13. For example, suppose that $\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal {X})$, $i \mapsto \mathcal{F}_ i$ is a diagram. Consider the object $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the pullback functor $x^*$ commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i$. Similarly we have $x^*\mathcal{F}|_{U_{\acute{e}tale}} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i|_{U_{\acute{e}tale}}$. Now by assumption each $x^*\mathcal{F}_ i|_{U_{\acute{e}tale}}$ is quasi-coherent, hence the colimit is quasi-coherent by the aforementioned Descent, Lemma 35.8.13. This proves (1).

It follows from (1) that cokernels exist in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ and agree with the cokernels computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ and let $\mathcal{K} = \mathop{\mathrm{Ker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. If we can show that $\mathcal{K}$ is a locally quasi-coherent module, then the proof of (2) is complete. To see this, note that kernels are computed in the category of presheaves (no sheafification necessary). Hence $\mathcal{K}|_{U_{\acute{e}tale}}$ is the kernel of the map $\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{G}|_{U_{\acute{e}tale}}$, i.e., is the kernel of a map of quasi-coherent sheaves on $U_{\acute{e}tale}$ whence quasi-coherent by Descent, Lemma 35.8.13. This proves (2).

Parts (3), (4), and (5) follow in exactly the same way. Details omitted. $\square$

In the generality discussed here the category of quasi-coherent sheaves is not abelian. See Examples, Section 108.13. Here is what we can prove without any further work.

Lemma 94.11.9. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.

  1. The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the categories $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$, $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, $\textit{Mod}(\mathcal{O}_\mathcal {X})$, and $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$.

  2. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

  3. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. Let $\mathcal{I} \to \mathit{QCoh}(\mathcal{O}_\mathcal {X})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Viewing $\mathcal{F}_ i$ as quasi-coherent modules in the Zariski topology (Lemma 94.11.4), we may consider the object $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the restriction functor $x^*$ (Section 94.9) commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i$ in $\textit{Mod}((\mathit{Sch}/U)_{Zar}, \mathcal{O})$. Observe that $x_ i^*\mathcal{F}_ i$ is a quasi-coherent object (because restrictions of quasi-coherent modules are quasi-coherent). Thus by the equivalence in Descent, Proposition 35.8.11 and by the compatibility with colimits in Descent, Lemma 35.8.13 we conclude that $x^*\mathcal{F}$ is quasi-coherent. Thus $\mathcal{F}$ is quasi-coherent, see Lemma 94.11.4. Thus we see that $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$. Since the other categories listed are full subcategories of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ we conclude part (1) holds.

Parts (2) and (3) are proved in the same way. Details omitted. $\square$

[1] This is nonstandard notation.

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